Sensitivity Analysis of the Temperature Field of Surrounding Rock in Cold-Region Tunnels Using a Fully Coupled Thermo-Hydrological Model

Abstract

The thermo-hydrological (TH) coupling model constitutes the foundational framework for investigating the temperature distribution of surrounding rock in cold region tunnels. In this study, a fully coupled TH model is proposed that takes into account multiple physical phenomena during the freezing process of surrounding rock. Firstly, the model was established based on thermodynamics, seepage theory, and ice–water phase change theory, which accounted for unfrozen water, latent heat of phase change, ice impedance, and convective heat transfer. The model was successfully verified by comparing its results to field data. Next, the sensitivity of surrounding rock temperature to environmental, thermodynamic, seepage, and coupling parameters in the fully coupled TH model was systematically studied using a numerical analysis method. The results show that the annual temperature amplitude and thermal conductivity represent the main factors affecting the surrounding rock temperature at a radial depth of 0 m, while the initial temperature and porosity are the key factors at a radial depth of 5 m. Permeability has the least influence on the surrounding rock temperature, but the temperature field will experience sudden changes if its value exceeds its value exceeds 1 × 10⁻¹² m². Finally, using the proposed numerical model, the thickness of insulation layer was simulated, and the degree of influence of the parameters on the thickness of insulation layer was analyzed. This study reveals that the annual temperature amplitude has the greatest influence on the calculation of insulation layer thickness, with its normalized sensitivity factor being approximately 50%. These findings not only expand the methodology for exploring the laws of TH coupling but also provide a theoretical foundation for improving the parameter calibration efficiency and calculation accuracy of the fully coupled TH model, and they have significant reference value.

1. Introduction

The permafrost area in China accounts for about 1/4 of the total land area of the country and is mainly located in the west and north. These areas have cold climates and complex natural conditions, and they are located at the country’s borders, making them areas of strategic importance [1,2]. With economic development and increased strategic defense requirements, the high-altitude western and high-latitude northern cold regions will see a surge in transport infrastructure construction and a large number of cold-zone tunnels will inevitably appear [3]. Cold-region tunnels often suffer from frost damage during construction and operation, such as hanging ice and frost cracks in the lining, which seriously affects the normal operation of the tunnel [4,5,6]. Accurate knowledge of the temperature field in cold-region tunnels is essential for effective implementation of anti-freezing measures. Therefore, it is imperative to comprehensively investigate the variation pattern and distribution characteristics of the surrounding rock temperature in order to develop effective prevention strategies [7,8]. A reasonable model and accurate parameters are a prerequisite for accurate envelope temperature calculations, so it is necessary to investigate models and parameters for temperature calculations in cold-region tunnels.

Numerous scholars worldwide have conducted extensive research on the calculation models and parameters for tunnels in cold regions. Tan et al. [9] established a heat transfer model for porous media with phase change based on the “unfrozen-freezing-frozen” theory and analyzed the effects of unfrozen water content and latent heat on the temperature distribution of surrounding rock. Zhou et al. [10] established a transient heat transfer model considering convective heat transfer and analyzed the influence of wind speed on the temperature field of the surrounding rock. Sun et al. [11] established a two-dimensional transient temperature field calculation model that did not consider the effects of groundwater seepage and phase change and analyzed the impact of annual average temperature, annual temperature amplitude, and initial ground temperature on the radial temperature field of the tunnel using a single-factor circulation method. Lai et al. [12] established a TH coupling model based on heat transfer and seepage theories and used the finite element method to investigate the freeze–thaw state of surrounding rock in cold-region tunnels. Zhang et al. [13] obtained the finite element formula considering the temperature field with phase change using the Galerkin method and investigated the characteristics of the temperature field of the surrounding rock in cold-region tunnels under different initial ground temperatures and thermal insulation layer thermal conductivities. These theoretical models contribute to understanding the freezing process of surrounding rocks in cold-region tunnels and provide reference for further research. However, important physical processes and phenomena such as ice–water phase change, convective heat transfer, ice impedance, and unfrozen water, which exist in the freeze–thaw process of surrounding rock in cold-region tunnels, are often ignored or cannot be comprehensively considered in research. Therefore, it is necessary to propose a comprehensive calculation model that considers various freezing and thawing physical processes and analyze the impact of parameters in the model on the temperature field of the surrounding rocks.

To quantitatively analyze the impact of model parameters on the surrounding rock temperature, many scholars have introduced the method of sensitivity analysis. Li et al. [14] combined numerical simulation and sensitivity analysis to study the influence of thermal parameters for surrounding rock on the temperature field of cold-region tunnels, and they concluded that the initial temperature of surrounding rock is the most sensitive parameter affecting the tunnel temperature field. Ma et al. [15] conducted a sensitivity analysis of energy tunnel parameters based on a TH coupling model and determined the relative influence of thermal parameters on the thermal performance of energy tunnels. Zhao et al. [16] proposed a calculation formula for determining the freezing depth of surrounding rock in cold-region tunnels and used the sensitivity analysis method to study the effects of different influencing parameters on the freezing depth, with the most sensitive parameter being the annual temperature amplitude. These research results have greatly enriched the application of sensitivity analysis in surrounding rock temperature fields. However, there is relatively little research on the sensitivity of rock temperature fields to seepage and coupling parameters, and there are few reports on the spatial distribution characteristics of sensitivity of temperature to multiple factors.

In summary, there has been relatively little research on the sensitivity of the surrounding rock temperature in cold-region tunnels to the parameters of a TH coupling model that considers the effects of ice–water phase change, convective heat transfer, ice impedance, and unfrozen water. In this paper, we first establish a fully coupled multidimensional TH model considering unfrozen water, latent heat of phase change, ice impedance, and convective heat transfer, based on thermodynamics, seepage theory, and ice–water phase change theory, and we validate the model using typical engineering scenarios. We then comprehensively analyze the sensitivity and spatial distribution characteristics of the surrounding rock temperature in cold-region tunnels based on the environmental parameters, thermal parameters, seepage parameters, and coupling parameters proposed in the numerical model. Finally, we successfully use sensitivity analysis of the TH coupling model parameters in the calculation of the insulation layer in practical engineering and discuss the main parameters that affect the calculation of the insulation layer thickness. The research results can provide a reference for the calculation of the surrounding rock temperature in cold-region tunnels.

2. A Fully Coupled Multidimensional TH Model

2.1. Governing Equations

In the freezing process, the convective heat transfer caused by the pore water seepage and the latent heat of phase change caused by the ice–water phase change are considered, in addition to considering the heat conduction between the lining and the surrounding rock. Assuming that the rock mass is in a saturated state, the governing equation for the temperature field is as follows [17]:

$${\left(\rho C_p\right)}_{eff}{\displaystyle \frac{\partial T}{\partial t}}+{\rho }_f{C}_{p,f}\overrightarrow{u}\nabla \mathrm{T}+\nabla ·\overrightarrow{q}=0$$

(1)

$$\overrightarrow{q}=-{\lambda }_{eff}\nabla T$$

(2)

where $\rho$ is the density of the surrounding rock (kg/m³); ${\left(\rho C_p\right)}_{eff}$ is the equivalent volume heat capacity (J/(m³·°C)); ${\lambda }_{eff}$ is the equivalent heat conductivity coefficient (W/(m·°C)); $\overrightarrow{q}$ is the heat flux density (W/m²); $\overrightarrow{u}$ is the relative velocity vector of water (m/s); $T$ is the surrounding rock temperature (°C); $C_{p,f}$ is the apparent heat capacity (J/m³·°C).

Frozen rock masses are composed of a particle skeleton, water, and ice. For low-temperature rock masses, the equivalent volume heat capacity and equivalent heat conductivity coefficient are both expressed as volumetrically weighted averages [18].

$${\left(\rho C_p\right)}_{eff}={\theta }_s{\rho }_s{C}_{p,s}+{{\theta }_w\rho }_w{C}_{p,w}+{\theta }_i{\rho }_i{C}_{p,i}$$

(3)

$${\lambda }_{eff}={\theta }_s{\lambda }_s+{\theta }_w{\lambda }_w+{\theta }_i{\lambda }_i$$

(4)

where

$${\theta }_s=1-\epsilon$$

(5)

$${\theta }_w=\epsilon S_w$$

(6)

$${\theta }_i=\epsilon \left(1-S_w\right)$$

(7)

where $C_p$ is the heat capacity at constant pressure (J/(kg·$°\mathrm{C}$)); $\lambda$ is the heat conductivity coefficient (W/(m·°C)); $\theta$ is the volume content; subscripts s, w, and I represent the rock skeleton, water, and ice, respectively; $\epsilon$ is porosity; $S_w$ is saturation, which is the volume ratio of liquid water to the pore.

There is still pore water in the pores, considering that the pore water in the rock mass does not completely freeze into ice under low-temperature conditions. The Heaviside function $H\left(t\right)$ is used to characterize the change in the volume of liquid water, which is expressed as follows:

$$S_w=S_{res}+\left(1-S_{res}\right)H\left(t\right)$$

(8)

where $S_{res}$ is residual saturation, that is, the volume ratio of unfrozen water to the pore at low temperatures.

By utilizing the apparent heat capacity method and simplifying the latent heat, the apparent heat capacity is proposed by [19]:

$$C_{p,f}={\displaystyle \frac{1}{{\rho }_w}}\left({\theta }_i{\rho }_i{C}_{p,i}+{\theta }_w{\rho }_w{C}_{p,w}\right)+L{\displaystyle \frac{\partial \frac{1}{2}\frac{{\theta }_i{\rho }_i-{\theta }_w{\rho }_w}{{\theta }_i{\rho }_i+{\theta }_w{\rho }_w}}{\partial T}}$$

(9)

The heat transfer differential equation considering the unfrozen water content, phase change, and fluid heat transfer effects can be obtained by substituting Equations (2)–(9) into Equation (1).

The ice–water flow equation in frozen rock can be expressed in the form of the continuity equation in terms of pressure [20,21].

$$S_w\epsilon {\rho }_w\beta {\displaystyle \frac{\partial p}{\partial t}}+\nabla \left({\rho }_w\overrightarrow{u}\right)+\epsilon \left({\rho }_w-{\rho }_i\right){\displaystyle \frac{\partial S_w}{\partial t}}=0$$

(10)

$$\overrightarrow{u}=-{\displaystyle \frac{k}{\eta }}\left(\nabla p+{\rho }_{w}g\right)$$

(11)

where $p$ is the pressure of water (Pa); $\beta$ is fluid compressibility (1/Pa); $k$ is permeability (m²); $\eta$ is the dynamic viscosity of water (Pa⋅s).

In the frozen rock mass, the ice volume content increases, occupying the pores and thus reducing the channels of water migration. Therefore, the impedance effect of pore ice under negative temperature conditions on permeability is considered, and the concept of relative permeability is proposed by [22]:

$$k_r=max\left\{{10}^{-\mathsf{\Omega }\epsilon S_i}\right.\left.,{10}^{-6}\right\}$$

(12)

where $\mathsf{\Omega }$ is the impedance factor; $k_r$ is the relative permeability (m²).

Thus, the permeability of the rock mass in the ice–water system can be expressed as follows:

$$k=k_r{k}_{int}$$

(13)

where $k_{int}$ is intrinsic permeability at a normal temperature (m²).

Equations (1) and (10) are the fully coupled multidimensional TH control equations for cold-region tunnels.

2.2. The Boundary and Initial Conditions

To solve the TH coupling Equations (1) and (10), the initial and boundary conditions are given as follows [23]:

(1)

Dirichlet boundary condition:

$$T=T_0,p=p_0$$

(14)

(2)

Neumann boundary condition:

$$-\overrightarrow{n}\cdot \overrightarrow{q}=q_0,-\overrightarrow{n}\cdot \overrightarrow{u}=u_0$$

(15)

(3)

Initial condition:

$${\left.T\right|}_{t=0}=T_1, {\left.p\right|}_{t=0}=p_1$$

(16)

After substituting the boundary and initial conditions into the TH coupling equations, the temperature distribution of the surrounding rock can be obtained. However, the proposed TH coupling model is highly nonlinear due to the time-varying and mutually interacting coefficients in the equations, making it difficult to obtain analytical solutions. Numerical methods can be employed to solve TH coupling equations. In this study, the temperature distribution of the surrounding rock was solved by coupling the heat transfer in porous media interface and the Darcy’s law interface in the numerical simulation software COMSOL Multiphysics(6.2).

2.3. Model Validation

Smith et al. [24] conducted model tests to observe and analyze the temperature field of silt around a gas pipeline in cold regions. Considering the typicality of this case, the established TH coupling model in this study was used to analyze the freezing problem in the Smith model test. By comparing with the actual measured temperature field around the gas pipeline, the accuracy of the model proposed in this paper was verified. The frozen pipeline model and conditions set by Smith are shown in Figure 1.

The latent heat of ice–water phase change is 333 kJ/kg. When the temperature is below −0.5 °C, the permeability of the silt is 1.28 × 10⁻¹² m², while when the temperature is above 0.5 °C, the permeability of the silt is 10⁻⁸ m². The values of other main physical parameters used in the calculation are listed in

Table 1. The main physical parameters in the calculation.

Item	Density $\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$	Thermal Conductivity Coefficient $\left(\mathbf{W}/\left(\mathbf{m}\cdot °\mathbf{C}\right)\right)$	Constant Pressure Heat Capacity $\left(\mathbf{J}/\left(\mathbf{k}\mathbf{g}\cdot °\mathbf{C}\right)\right)$
Silt	2650	1.7	900
Water	1000	0.56	4180
Ice	917	2.24	2100

.

Smith obtained the frozen depths around pipelines at different times through model experiments. To rigorously validate the physical completeness of our fully coupled thermal–hydrological (TH) model, which explicitly resolves ice–water phase change, unfrozen water dynamics, and ice impedance effects, the freezing depths under the same conditions were calculated. As shown in Figure 2, the measured results and the calculated results from the numerical simulation are basically the same, with a maximum error of only 4 cm. Quantitative evaluation metrics demonstrate excellent model agreement, with a very high coefficient of determination (R² = 0.998) and a low prediction error (RMSE = 1.48 cm), further confirming the model’s validity, which proves the correctness of the fully coupled multidimensional TH coupling model established in this paper.

3. Parameter Sensitivity Analysis for Fully Coupled TH Model in Cold-Region Tunnels

3.1. Numerical Model and Parameter Range

A calculation model for cold-region tunnels (Figure 3) was constructed based on the TH coupling model. Due to the symmetry of the tunnel section, the right half was selected for research and analysis. The thickness of the lining (preliminary lining and secondary lining) is 80 cm. The upper boundary condition has minimal impact on the temperature distribution of shallow zones (≤20 m) in the surrounding rock temperature field [25], effectively eliminating thermal boundary effects in near-surface regions. Thus, the tunnel burial depth is set to 21 m in this study.

The heat boundary conditions are as follows. The lateral boundaries AG and ED are symmetrical, and the lateral boundary BC is adiabatic. Heat flux at boundary DC is 0.01 W/m². The variation in atmospheric temperature shows periodic changes, and natural surfaces AB and GEF can be expressed in the form of a sine function (Equation (17)). The convective heat transfer coefficient between air and surrounding rock is α = 15 W/(m²⋅°C).

$$T_a=T_0+A\mathrm{s}\mathrm{i}\mathrm{n}(\omega t+\phi )$$

(17)

where $T_a$ is the temperature inside the tunnel (°C); T₀ is the annual average temperature of the air inside the tunnel (°C); A is the amplitude of the air temperature inside the tunnel (°C); ω is the period of temperature variation, typically taken as 1 year; φ is the initial phase, determining the starting moment.

The water boundary conditions are as follows. The boundaries AG and ED are symmetrical. The lateral boundary CD is a waterproof boundary. The boundary BC and the point F are pressure boundaries. According to relevant studies [16,22,26,27], the range of TH coupling calculation parameters for tunnels in cold regions are summarized in

Table 2. Calculation parameters and value range of TH coupling model.

Number	Parameter	Value	Paper
1	${\lambda }_s/\left(\mathrm{W}/\left(\mathrm{m}\cdot \mathrm{K}\right)\right)$	2.50~3.20	[16]
2	${\lambda }_w/\left(\mathrm{W}/\left(\mathrm{m}\cdot \mathrm{K}\right)\right)$	0.56	[16]
3	${\lambda }_i/\left(\mathrm{W}/\left(\mathrm{m}\cdot \mathrm{K}\right)\right)$	2.14	[16]
4	${\lambda }_c/\left(\mathrm{W}/\left(\mathrm{m}\cdot \mathrm{K}\right)\right)$	1.85	[25]
5	$C_{p,s}/\left(\mathrm{J}/\left(\mathrm{k}\mathrm{g}\cdot \mathrm{K}\right)\right)$	750~1200	[16]
6	$C_{p,w}/\left(\mathrm{J}/\left(\mathrm{k}\mathrm{g}\cdot \mathrm{K}\right)\right)$	4180	[16]
7	$C_{p,i}/\left(\mathrm{J}/\left(\mathrm{k}\mathrm{g}\cdot \mathrm{K}\right)\right)$	2100	[16]
8	$C_{p,c}/\left(\mathrm{J}/\left(\mathrm{k}\mathrm{g}\cdot \mathrm{K}\right)\right)$	1455	[16]
9	${\rho }_s/\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)$	2254~2650	[16]
10	${\rho }_w/\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)$	1000	[16]
11	${\rho }_i/\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)$	917	[16]
12	${\rho }_c/\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)$	2500	[26]
13	$\epsilon /\%$	2~40	[16]
14	$S_{res}/\%$	5~30	[22]
15	$T_O/\left(°\mathrm{C}\right)$	0~10	[26]
16	$A/\left(°\mathrm{C}\right)$	10~26	[26]
17	$b/\left(°\mathrm{C}\right)$	2~10	[26]
18	$k_{int}/\left({\mathrm{m}}^2\right)$	1 × 10−10~9 × 10−15	[22]
19	$L/\left(\mathrm{k}\mathrm{J}/\mathrm{k}\mathrm{g}\right)$	334	[26]
20	$\Omega$	50	[27]
21	$T_{Pc}/\left(°\mathrm{C}\right)$	−0.5	[27]
22	$\mathsf{\Delta }T/\left(°\mathrm{C}\right)$	1	[27]

Note: Influence of groundwater head and permeability coefficient on temperature field is reflected in convection heat transfer effect of groundwater seepage velocity on temperature field. Therefore, in this paper, groundwater head remains constant, and different seepage velocities are obtained by changing permeability coefficient.

.

3.2. Method for Sensitivity Analysis

Through the analysis of the parameters in the TH coupling model for cold-region tunnels, it can be seen that 22 parameters are required to accurately calculate the temperature and seepage fields. Among them, 9 parameters need to be determined according to the actual situation of the tunnel. The 9 factors that affect the surrounding rock temperature field are classified into four categories: environmental parameters, including the annual average temperature $T_O$ of the air, the annual temperature amplitude $\mathrm{A}$ of the air, and the initial temperature $b$ of the surrounding rock; thermodynamic parameters of the surrounding rock, including the density ${\rho }_s$ of the skeleton particles, thermal conductivity ${\lambda }_s$, and specific heat $C_{p,s}$; seepage parameters of the surrounding rock, including permeability $k_{int}$; and coupling parameters, including porosity $\epsilon$ and residual saturation degree $S_{res}$, both of which can affect the temperature and seepage fields. Considering that different parameters have different degrees of influence on the numerical calculation results, sensitivity analysis of the parameters is required.

Sensitivity analysis [28] is a method for analyzing system stability and evaluating the degree of influence of different factors on the model results. Firstly, it is necessary to establish a system model and clarify the functional relationship between the influencing factors and the system characteristics, that is, $y=f\left(x_1,x_2,\cdots {,x}_n\right)$, where y is the system characteristic value, and $x=\left(x_1,x_2,\cdots {,x}_n\right)$ is the set of influencing factors of the model. For complex system functional relationships that are difficult to express, numerical methods can be used for representation. Next, after establishing the system model, a standard parameter set and standard system characteristic values are provided; that is, at the standard parameter values $x^{*}=\left({x_1}^{*},{x_2}^{*},\cdots ,{x_n}^{*}\right)$, the standard system characteristic value is $y^{*}=f\left({x_1}^{*},{x_2}^{*},\cdots ,{x_n}^{*}\right)$. When analyzing the influence of parameter $x_k$ on characteristic $y$, single-factor analysis can be used. The values of other parameters are kept constant at their standard values, and $x_n$ is allowed to vary within its permissible range, so the change in the system characteristic value is expressed as $∆y=f\left({x_1}^{*},{x_2}^{*},\cdots ,{{x_k}^{*}+∆x}_k\cdots ,{x_n}^{*}\right)-f\left({x_1}^{*},{x_2}^{*},\cdots ,{x_n}^{*}\right)$. If a small change in $x_k$ leads to a large change in the system characteristic value, it indicates that the system characteristic value is highly sensitive to this parameter. In order to compare and analyze parameters with different units, nondimensional sensitivity is used for representation.

$$S_k=\left|{\displaystyle \frac{{x_k}^{*}}{y^{*}}}\times {\displaystyle \frac{∆y}{{∆x}_k}}\right|$$

(18)

where $S_k$ is the sensitivity to $x_n$ of the system characteristic.

When analyzing multiple factors, it is necessary to normalize the sensitivity for comprehensive analysis. Specifically, we make the sum of sensitivity of all influencing parameters for the same system characteristic equal to 1 [14]. The normalized sensitivity can be obtained:

$${S_k}^{\prime }={\displaystyle \frac{S_k}{\sum _{i=1}^n{S}_i}}$$

(19)

where ${S_k}^{\prime }$ is the normalized sensitivity to $x_n$ of the system characteristic.

According to the sensitivity analysis methods mentioned above, the parameters that affect the system include $T_O$, $\mathrm{A}$, $b$, ${\rho }_s$, ${\lambda }_s$, $C_{p,s}$, $k_{int}$, $\epsilon$, and $S_{res}$. The minimum temperature during a calculation period is taken as the standard characteristic value. Table 2 is used to determine the range of variation for each of the 9 parameters, and the standard parameter set and the parameter variation range are determined accordingly, as shown in

Table 3. Standard parameter set and parameter variation range.

Condition	${\mathit{T}}_{\mathit{O}}\left(°\mathbf{C}\right)$	$\mathbf{A}\left(°\mathbf{C}\right)$	$\mathit{b}\left(°\mathbf{C}\right)$	${\mathit{C}}_{\mathit{p},\mathit{s}}$ $\left(\mathbf{J}/\left(\mathbf{k}\mathbf{g}\cdot °\mathbf{C}\right)\right)$	${\mathit{\lambda }}_{\mathit{s}}$ $\left(\mathbf{W}/\left(\mathbf{m}\cdot °\mathbf{C}\right)\right)$	${\mathit{\rho }}_{\mathit{s}}$ $\left(\mathbf{k}\mathbf{g}/{\mathbf{m}}^3\right)$	${\mathit{k}}_{\mathit{i}\mathit{n}\mathit{t}}$ $\left({\mathbf{m}}^2\right)$	$\mathit{\epsilon }$	${\mathit{S}}_{\mathit{r}\mathit{e}\mathit{s}}$
1	0, 2, 4, 6, 8	26	2	950	3	2550	1 × 10−14	0.3	0.1
2	0.8	10, 15, 20, 25, 30	2	950	3	2550	1 × 10−14	0.3	0.1
3	0.8	26	2, 4, 6, 8, 10	950	3	2550	1 × 10−14	0.3	0.1
4	0.8	26					1 × 10−12, 1 × 10−13, 1 × 10−14, 1 × 10−15
5	0.8	26				2250, 2350, 2450, 2550, 2650
6	0.8	26			2.4, 2.6, 2.8, 3, 3.2
7	0.8	26		750, 850, 950, 1050, 1150
8	0.8	26						0.15, 0.20, 0.25, 0.30, 0.35
9	0.8	26							0.05, 0.10, 0.15, 0.20, 0.25

in this paper.

3.3. Sensitivity Analysis of Temperature Field to Different Influencing Factors

3.3.1. Influence of Different Parameters on Temperature Field

Analyzing nine parameters’ effects on the temperature at the radial depth of 0 m in the surrounding rock using the tunnel arch as an example, the results are shown in Figure 4. The annual mean temperature, initial rock temperature, density, thermal conductivity, heat capacity, and porosity are positively correlated with the rock temperature. Meanwhile, the annual temperature amplitude and residual saturation are negatively correlated with the temperature. The permeability coefficient exhibits a distinct pattern compared to other factors; when the permeability coefficient exceeds 1 × 10⁻¹³ m², the temperature field experiences a sudden change (Figure 4h).

Here has been considerable research on the effects of environmental parameters such as annual mean temperature, annual temperature amplitude, and initial rock temperature, as well as thermodynamic parameters of the surrounding rock, including density, thermal conductivity, and heat capacity, on the temperature field of surrounding rock in cold-region tunnels, which we will not elaborate on here. Instead, the focus will be on analyzing the effects of permeability, porosity, and residual saturation on the surrounding rock temperature field.

When the temperature drops below the freezing point of the rock, some of the pore water in the rock freezes into ice. Due to the effect of the surface energy of rock particles, there is always a layer of film water (unfrozen water) on the surface. Therefore, the porosity $\epsilon$ and residual saturation $S_{res}$ affect the composition of the rock mass, resulting in changes in the equivalent thermal conductivity, equivalent heat capacity, permeability, and latent heat released by the rock mass. Figure 4h–i show the temperature changes at the radial depth of 0 m in the surrounding rock of the arch at different porosity and residual saturation values. When the porosity of the surrounding rock increases and the residual moisture content decreases, there will be less unfrozen water at low temperatures, and more liquid water will freeze into ice and release latent heat. At the same time, the increase in ice volume reduces the permeability coefficient, causing a weaker convective heat transfer effect and resulting in an increase in rock temperature.

TAs shown in Figure 4, the permeability coefficient exhibits distinct behavior compared to the other parameters, and its variation induces abrupt changes in the surrounding rock temperature. Consequently, this study focuses on analyzing the influence of the permeability coefficient on the temperature field distribution within the surrounding rock, as depicted in Figure 5. The gray line represents the 0 °C isotherm of the surrounding rock. The temperature distributions in Figure 5b–d are relatively consistent and all lower than that in Figure 5a. In Figure 5a, the freezing front largely conforms to the tunnel profile, exceeding it only slightly at the vault and inverted arch. As the permeability coefficient decreases (Figure 5b–d), the extent of the freezing front expands significantly. This phenomenon occurs because an increase in the permeability coefficient accelerates the seepage velocity of water in the surrounding rock. This accelerated flow, on one hand, transports heat from distal regions of the rock mass into the tunnel, and on the other hand, it intensifies convective heat transfer. Consequently, the surrounding rock becomes more resistant to freezing.

The influence of permeability on the temperature field is essentially the influence of seepage velocity on the temperature field. Keeping the groundwater table constant, the permeability of the surrounding rock is changed and the minimum temperature of the surrounding rock at different depths in a tunnel during one period is calculated using a TH coupling model. The results are shown in Figure 6. When the permeability of the surrounding rock is 1 × 10⁻¹² m², 1 × 10⁻¹³ m², 1 × 10⁻¹⁴ m², and 1 × 10⁻¹⁵ m², respectively, the seepage velocity of groundwater at normal temperature in the surrounding rock is 3 × 10⁻⁶ m/s, 3 × 10⁻⁷ m/s, 3 × 10⁻⁸ m/s, and 3 × 10⁻⁹ m/s, respectively. As can be seen from Figure 5, when the seepage velocity is less than 3 × 10⁻⁶ m/s, the temperature variation in the surrounding rock at the same position is not significant, and the change in seepage velocity has no obvious effect on the temperature field of the surrounding rock. When the seepage velocity is 3 × 10⁻⁶ m/s, the temperature field changes significantly, and the convective heat transfer effect of the fluid is significant at this time. When the seepage velocity is 3 × 10⁻⁶ m/s, the temperature at a depth of 1 m or more is 2 °C, and the temperature does not change with the change in air temperature. This is because the initial temperature of the surrounding rock is 2 °C, and the groundwater discharged from the drainage outlet far from the tunnel is equivalent to a constant heat supply source continuously heating the surrounding rock of the tunnel. The greater the seepage velocity, the more heat is input per unit time, which can offset the influence of the air temperature change on the temperature field of the surrounding rock. It can be seen from the above analysis that the sensitivity factor of permeability is mainly due to the sudden change in temperature field caused by the critical seepage velocity.

There exist temperature differences at different locations of the same depth for surrounding rock with the same permeability. It can be seen that the surrounding rock temperature at the vault and arch foot differs little and is far lower than the temperature at the inverted arch form in Figure 6. Firstly, this is because the distance between the surrounding rock at the vault and arch foot and the outside air is only 0.8 m (lining), while the distance between the surrounding rock at the inverted arch and the outside air is 2.4 m (lining + pavement), which causes more energy to be consumed when it reaches the temperature measuring point at the inverted arch. Secondly, this is because the temperature measuring point at the inverted arch is closest to the drainage outlet, which results in the maximum flow rate of pore water and a larger convective heat transfer effect.

3.3.2. The Spatial Distribution Characteristics of Normalized Sensitivity Factors

As the tunnel does not have a standard circular section and is affected by the concrete pavement, the temperature distribution in different parts of the tunnel is not the same. Therefore, according to Equation (19), the normalized sensitivity factors of nine parameters including $T_O$, $\mathrm{A}$, $b$, ${\rho }_s$, ${\lambda }_s$, $C_{p,s}$, $k_{int}$, $\epsilon$, and $S_{res}$ were performed at radial depths of 0 m, 1 m, 2 m, 3 m, and 4 m from the lining surface at the vault, arch foot, and inverted arch of the tunnel. The results are shown in Figure 7, and it can be seen that the normalized sensitivity factors of the annual mean temperature, permeability, and residual saturation at different depths and locations are all less than 5%, which are insensitive parameters. However, the normalized sensitivity factors of the annual temperature amplitude, initial temperature, and porosity are all greater than 30%, which are sensitive parameters.

The distribution of normalized sensitivity factors for different parameters is shown in Figure 8. It can be seen that there are significant differences in sensitivity at different depths. At a radial depth of 0 m, the annual temperature amplitude and thermal conductivity are the most sensitive factors; at radial depths of 1 and 2 m, the annual temperature amplitude and porosity are the most sensitive factors; at radial depths of 3 and 4 m, the initial temperature and porosity are the most sensitive factors. Within the radial depth range of 2 m of the surrounding rock, the distribution characteristics of normalized sensitivity factors for temperature at the vault and arch foot are the same. The sensitivity of environmental parameters decreases with increasing depth, while the sensitivity of thermodynamic parameters remains almost constant and the sensitivity of coupled parameters increases with increasing depth. Outside the radial depth range of 2 m of the surrounding rock, the sensitivity of environmental parameters increases with increasing depth, the sensitivity of thermodynamic parameters decreases with increasing depth, and the sensitivity of coupled parameters remains almost constant. This is because the asymmetry of the tunnel structure drainage path leads to different distributions of normalized sensitivity factors for different parameters in space.

4. The Application of Sensitivity Analysis of TH Coupling Model Parameters

The distribution of the temperature field in cold-region tunnels is an important theoretical basis for anti-freezing and insulation design. In this regard, the temperature at the interface between preliminary lining and surrounding rock above 0 °C is taken as the reference basis for insulation layer design. Therefore, analyzing the sensitivity of model parameters to the temperature field 0 m away from the preliminary lining and determining the important influencing parameters in insulation layer design are crucial.

Figure 9 shows the sensitivity of each parameter in insulation layer design, and it can be observed that the sensitivity varies in different locations, but the order of the parameters’ influence on the temperature field remains unchanged. The variation in annual temperature amplitude is the most sensitive factor, with a normalized sensitivity factor of 42.52% to 58.61%, followed by the thermal conductivity coefficient, porosity, density, heat capacity, residual saturation, initial temperature, permeability, and annual average temperature, with a total normalized sensitivity factor of 24.52% to 37.29%. The residual saturation, initial temperature, permeability, and annual average temperature have relatively small impacts on the temperature field distribution, with a normalized sensitivity factor of about 2% each. Therefore, the high-sensitivity parameter is the annual temperature amplitude in insulation layer design, while the low-sensitivity parameters are the porosity, density, heat capacity, residual saturation, initial temperature, permeability, and annual average temperature.

The insulation layer is laid on the surface (Figure 9), and the calculation parameters are based on the data in Table 3. Taking tunnel in Figure 3 as an example, the insulation layer thickness was determined through parameter scanning analysis: temperature distributions under multiple thickness scenarios were systematically computed, with continuous monitoring at surrounding rock–lining interfaces. The critical thickness was identified when all monitored regions maintained a temperature greater than 0 °C. The insulation layer is made of polyurethane insulation board, with a thermal conductivity of 0.027 W/(m⋅°C), a density of 40 kg/m³, and a specific heat of 5000 J/(kg⋅°C). The insulation layer thickness is calculated with standard parameters, and then the thickness of the insulation layer is calculated when seven low-sensitivity parameters simultaneously produce errors and one high-sensitivity parameter produces an error. To prevent the effects of errors in the seven low-sensitivity parameters from canceling each other out, we used the conclusion from Figure 4 that residual saturation and annual temperature amplitude are negatively correlated with the temperature field, while the remaining parameters are positively correlated with the temperature field. The specific values and results of the calculation parameters are shown in

Table 4. Insulation layer thickness under different parameter errors.

	Standard Parameters	Highly Sensitive Parameters with 10% Error	Low Sensitive Parameters with 10% Error	Highly Sensitive Parameters with 20% Error	Low Sensitive Parameters with 20% Error
$\epsilon$	0.3	0.3	0.33	0.3	0.36
${\rho }_s/\left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right)$	2550	2550	2805	2550	3060
$C_{p,s}$$/\left(\mathrm{J}/\left(\mathrm{k}\mathrm{g}\cdot \mathrm{K}\right)\right)$	950	950	1045	950	1140
$S_{res}$	0.1	0.1	0.09	0.1	0.08
$b$$/\left(°\mathrm{C}\right)$	2	2	2.2	2	2.4
$T_O/\left(°\mathrm{C}\right)$	0.8	0.8	0.88	0.8	0.96
$k_{int}/\left({\mathrm{m}}^2\right)$	1 × 10−14	1 × 10−14	1.1 × 10−14	1 × 10−14	1.2 × 10−14
$\mathrm{A}/\left(°\mathrm{C}\right)$	26	23.4	26	20.8	26
Thickness of Insulation layer/cm	9	8	8.5	7.2	8.1

. From Table 4, it can be seen that the thickness of the insulation layer calculated with a 10% error in one high-sensitivity parameter and a 10% error in seven low-sensitivity parameters is 8 cm and 8.5 cm, respectively, with errors of 1 cm and 0.5 cm, respectively. When seven low-sensitivity parameters all have a 20% error, the calculated thickness of the insulation layer is 8.1 cm, with an error of 0.9 cm, which is still smaller than the case of one high-sensitivity parameter having an error. Therefore, the accuracy of the most sensitive parameter (annual temperature amplitude) in the calculated results should be further improved in practical insulation layer design, while the parameters with lower sensitivity (porosity, density, heat capacity, residual saturation, initial temperature, permeability, and annual average temperature) can be estimated based on experience and relevant literature to simplify the calculation process.

5. Conclusions

This article presents a fully coupled TH model for surrounding rocks in cold-region tunnels that considers various physical phenomena during the freeze–thaw process. Sensitivity analysis methods are also introduced to quantitatively study the factors affecting the temperature field of the surrounding rocks. The analysis of the influence of permeability on the temperature and the spatial distribution characteristics of sensitivity factors are novel aspects in the framework of cold-region tunnels compared to previous works in the literature. The main conclusions are as follows:

(1)

Residual saturation, apparent heat capacity, and relative permeability are introduced to derive a TH coupling equation that considers various physical phenomena such as unfrozen water, latent heat of phase change, ice impedance, and convective heat transfer. These equations provide a more comprehensive and realistic approach to studying the fully coupled TH model of surrounding rocks in cold-region tunnels. The numerical results of the frozen depth obtained from this model are consistent with the model test results, demonstrating its high accuracy.

(2)

The annual temperature amplitude, initial ground temperature, and porosity exhibit high sensitivity in the TH model. In contrast, the annual average temperature and residual water saturation demonstrate lower sensitivity. Crucially, parameter sensitivity displays significant spatial dependence. At 0 m radial depth (tunnel lining interface), the annual temperature amplitude and thermal conductivity dominate temperature variations. At 1–2 m depth, temperature sensitivity is primarily governed by the annual temperature amplitude and porosity. At 3–4 m depth, the initial ground temperature and porosity emerge as the most influential parameters.

(3)

Permeability is identified as an insensitive parameter for seepage velocities below the critical threshold of 3 × 10⁻⁶ m/s (corresponding to permeability < 1 × 10⁻¹² m²). Within this range, variations in seepage velocity exert negligible influence on the surrounding rock’s temperature field. Notably, a critical transition occurs at seepage velocities ≥ 3 × 10⁻⁶ m/s, where the temperature field exhibits an abrupt shift due to enhanced convective heat transfer by pore fluid.

(4)

The annual temperature amplitude is a highly sensitive parameter affecting the thickness of the insulation layer, with a normalized sensitivity factor of 42.52% to 58.61%. Low-sensitivity parameters include porosity, density, specific heat capacity, residual moisture content, initial temperature, permeability, and annual average temperature, with a total normalized sensitivity factor of 24.52% to 37.29%. The normalized sensitivity factor for residual saturation, initial temperature, permeability, and annual average temperature is only about 2%.